On Deterministic and Stochastic Models of Kleptoparasitism By: M. Crowe, M. Fitzgerald, D. L. Remington, J. Rychtář Crowe, M., M. Fitzgerald, D.L. Remington, and J. Rychtář. 2009. On deterministic and stochastic models of kleptoparasitism. Journal of Interdisciplinary Mathematics 12:161-180 This is an Accepted Manuscript of an article published by Taylor & Francis Group in the Journal of Interdisciplinary Mathematics on May 28 2013 , available online at: http://www.tandfonline.com/10.1080/09720502.2009.10700620 . ***© Taylor & Francis & Taru Publications. Reprinted with permission. No further reproduction is authorized without written permission from Taylor & Francis. This version of the document is not the version of record. Figures and/or pictures may be missing from this format of the document. *** Abstract: Kleptoparasitism, the stealing of food items, is a common biological phenomenon that has been studied mostly with the help of deterministic dynamics for infinite populations. The infinite population assumption takes the models far from the biological reality. In this paper we provide a review of the main theoretical works on kleptoparasitism and then focus on the stochastic dynamics of kleptoparasitic individuals in finite populations. We solve the dynamics analytically for populations of 2 and 3 individuals. With the help of numerical solution of the dynamics, we were able to conclude that the behavior of the uptake rate in the population is mostly determined by the uptake rates at populations of 2 and 3 individuals. If the individuals do better in a pair, then the uptake rate is a decreasing function of the population size. If the individuals do better in a triplet than in a pair, then the uptake rate is a zigzag function with lows for even population sizes and ups for uneven population sizes. Keywords: Kleptoparasitism | game theory | strategy | finite populations Article: 1. Introduction Kleptoparasitism is a behavior in which one organism steals food or other resources that another organism has caught, killed, or otherwise prepared for its own use. This process occurs across a great diversity of taxa, with recent observations from large carnivorous mammals [11], seabirds [13], scavenging bird guilds [3], insects [23], fish [14], lizards [12], snails [17] and spiders [27, 28, 1, 18]. When considering kleptoparasitism we must ask, why would a species evolve this behavior? From an evolutionary standpoint the answer to this is quite logical: in order for a behavior to evolve in a population through natural selection, the individuals who follow this behavior must enjoy greater reproductive success (higher fitness) compared to others in the population, and this trait must be passed on to their future offspring. Since not all species that seem capable of kleptoparasitism show it, and since there is strong variation between and within species in the extent to which this tactic is used, there is a need for a predictive theoretical basis to explain this variation. Hence there has been a considerable body of theory aimed a predicting the evolutionarily stable use of kleptoparasitism in different ecological circumstances. In this paper we first review seminal theoretical papers focused on intraspecific kleptoparasitism, and then study the stochastic model introduced in [29] with a special focus on small population sizes. Almost all models developed to date are deterministic, based on a certain system of ordinary differential equations (ODEs). Consequently, these models assume a very large population size. On the other hand, the majority of model organisms that exhibit kleptoparasitism are birds, mammals and fish, with relatively small populations that are likely to violate assumptions of infinite population size models. To our knowledge, only two papers to date [22] and [29] proposed stochastic models for finite and thus potentially small populations. Moreover, [22] deal with a model that differs from the mainstream deterministic models and [29] focuses on situation for medium population sizes. Hence, models that can accurately describe kleptoparasitism in smaller population sizes are needed. 2. Deterministic models of kleptoparasitism In this section, we provide an overview of the main models and summarize the major assumptions and conclusions of each. 2.1 Holmgren, 1995 One of the first models focused on intraspecific kleptoparasitism was developed in [16] and it used foraging interference as its basis. In this model, an individual’s behavior was deterministic, always fighting whenever an individual encountered another individual who had a food item. The model also had individuals encountered each other randomly but at a constant rate. Because individuals were distinct, it allowed for one to examine the effect of a given type of individuals who differed in their dominance over each other. But the model was very complex and only allowed for a numerical, but not analytical, solution. 2.2 Stillman et al., 1997 In [26], the authors extended the existing behavior-based foraging interference models to kleptoparasitism. In their model, an individual could engage in one of four behaviors: searching, handling, fighting or avoiding. Their model had three key components: • the response of an individual to a competitor was not fixed, • individuals did not move randomly, and • that time was not the only currency by introducing prey as another form of currency. The authors applied the model to data collected on foraging shorebirds including oystercatchers, curlew, and knots. They found the model to be robust, accurately predicting interference 85% of the time. The authors suggested that future models should incorporate the idea that an individual follows optimal decision rules when deciding their behavior. Two major limitations of this paper include the assumptions that prey items were identical, and that the handler always defended the prey. 2.3 Ruxton and Moody, 1997 In [25], authors took the advice given in [26] and used a functional response model to bridge the gap between an individual’s behavior and the evolution of a population. Their approach allowed for an analytical solution of the model, not just a numerical one. In this model, individuals in the population were engaged in one of the following behaviors: • searching for a food item, • handling a food item, ² winning a contest over a food item, • losing a contest over a food item. The fight over a food item resulted when a searcher found a handler. The model examines an individual’s behavior over varying densities of prey and competitors and when there is more than one patch of food. It predicts that the proportion of individuals wanting to forage in the best patch will always be greater than the proportion of prey available. It also predicts that an individual’s decision to forage in a given patch is dependent on two things: the proportion of food in that patch and the distribution of food between other patches in the area. Two major limitations of this model were that: (1) individuals have equal competitive ability, and (2) any time a forager encountered an individual handling a prey item, it would attempt kleptoparasitism. 2.4 Broom and Ruxton, 1998 In [9], authors take a different tack than [25]. They divide the individuals in the population into three mutually exclusive groups: • searching for a food item or for a handler, • handling a food item, • fighting over the food item. They assumed that individuals can adopt different strategies (to steal or not to steal) and they examined what strategies can be evolutionarily stable. An evolutionary stable strategy (ESS) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy, [20]. The situation can be described by the diagram on Figure 1. It should be noted that the model assumes that a food is a shell type (see Section 2.6), i.e., handled for an exponentially distributed time; and that the fights last for an exponentially distributed time. There are two main advantages to assuming the exponentially distributed time for handling. (1) One can use ODEs in the model, (2) The food item has always the same value until eaten. The latter is a consequence of the fact that the conditional expectation for an exponentially distributed random variable is the same as the original expectations. Once a handling starts, the expected time to finish handling is defined as Th . Provided the individual did not finish handling in the time t0 , then at time t0 , the expected time to finish handling is again Th . This means, in particular, that the individuals can fight over the food item as long and as many times as they choose, yet the winner always get the same food item as it would get by finding it by itself. Figure 1. Schematics of the deterministic dynamics model Using a functional response framework, the model predicts an individual’s behavior under varying prey densities. If food is plentiful, the model predicts that no kleptoparasitism will take place. Once the food density drops below a critical threshold, kleptoparasitism will occur whenever the opportunity arises. The model assumes that the behavior of an individual is deterministic; individuals either always steal or never steal, and which strategy they employ depends on either food availability or the time (length) of the encounter. The model predicts that individuals will cross a threshold level where stealing is optimal when fights are not costly and when food is hard to find. One peripheral assumption of the model, that a food item requires a specific handling time prior to receiving the energy reward, is further explored in subsequent models. The paper [9] is a foundation for a vast majority of subsequent papers. 2.5 Ruxton and Broom, 1999 The model in [24] examined the trade off between investing time (or energy) in finding food on their own or by kleptoparasitizing from others. Specifically, they examined the assumption that the time of a contest over a food item is fixed.